Use the rules to find derivatives of the following functions at the specified values. a. f(x) = 2x³ at x = 2 f' (2)= g(z) =13x ½ at x = 3 g' (3)= h(z)= hx at x = 4 j(z) 130x¯¹ at x = 5 j' (5)=

After depreciating for 3 years at a rate of 16% per year, the computer is worth approximately $788.26.

To find the worth of the computer after depreciating for 3 years at a rate of 16% per year, we can use the formula for compound interest with depreciation.

Given:

Initial value (cost of the computer) = $1,495

Depreciation rate = 16% per year

Number of years = 3

1. Convert the depreciation rate to a decimal: 16% = 0.16.

2. Calculate the depreciation factor, which is (1 - depreciation rate):

Depreciation factor = 1 - 0.16 = 0.84.

3. Apply the formula for compound interest with depreciation:

Worth = Initial value * (Depreciation factor)^(Number of years).

Substituting the given values into the formula:

Worth = $1,495 * (0.84)^3.

Calculating the exponent:

Worth = $1,495 * 0.84 * 0.84 * 0.84.

Simplifying the expression:

Worth ≈ $788.26.

Therefore, after depreciating for 3 years at a rate of 16% per year, the computer is worth approximately $788.26.

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Answer: C

Step-by-step explanation:

since slope is rise/run its 2 and since the line is a negative slope the slope of the line is -2. and the y-intercept of the line is -2.

y = mx+b

m = -2

b= -2

Answer: C. y= -2x -2

The ordered pair (s,t) that satisfies the system are (1/2,4) and (1,5).

We can use the second equation to solve for one of the variables in terms of the other. Let's solve for t:

3t - 6s = 9

3t = 6s + 9

t = (2s + 3)

Now we can substitute this expression for t into the first equation and solve for s:

s/2 + 5/t = 3

s/2 + 5/(2s + 3) = 3

Multiplying both sides by the denominator (2s + 3) gives:

s(2s + 3)/2 + 5 = 3(2s + 3)

Simplifying and collecting like terms yields:

2s^2 + 3s + 10 = 6s + 9

2s^2 - 3s + 1 = 0

This quadratic equation can be factored as:

(2s - 1)(s - 1) = 0

So s = 1/2 or s = 1.

Now we can substitute these values for s into the equation we derived for t:

If s=1/2 then t=2(1/2)+3=4

If s=1 then t=2(1)+3=5

Therefore, the ordered pairs that satisfy the system are (1/2,4) and (1,5).

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Using a calculator, we can evaluate ln 0.732 to four decimal places. The correct answer is option D, -0.4719.

The natural logarithm of a number is the logarithm to the base e (approximately 2.71828), and ln 0.732 is the natural logarithm of the number 0.732.

To find the value of ln 0.732, we simply input the number into the calculator and hit the ln key.

The result is approximately -0.4719, rounded to four decimal places. Therefore, the correct answer is D, -0.4719.

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The required solution to the given equation for x using square roots is x = 5 or x = -1.

The quadratic equation is given as follows:

12-(x-2)² = 3

To solve the equation 12-(x-2)² = 3 using square roots, we need to isolate the squared term on one side of the equation and then take the square root of both sides.

We have:

12-(x-2)² = 3

-(x-2)² = -9

(x-2)² = 9

Apply the square root property in the equation,

x-2 = ±√9

x-2 = ±3

x = 2 ± 3

x = 5 or x = -1

Therefore, the solution for x using square roots is x = 5 or x = -1.

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a) The critical points are x = -1 and y = nπ, where n is an integer. The linearization of the system near the center is x' = ky, y' = -kx

b) The critical points are (1, 1) and (-1, -1). The linearization of the system near the degenerate critical point is x' = y, y' = 2x + 2y

(a) To find the critical points of the system, we need to solve the equations:

1 + x = 0 and sin y = 0 for x and y, respectively. This gives us x = -1 and y = nπ, where n is an integer. We can also find the linearizations near each critical point by computing the Jacobian matrix:

J(x, y) = [cos y, (1 + x)cos y - sin y; -1, sin y]

At the critical point (-1, nπ), the Jacobian matrix becomes:

J(-1, nπ) = [(-1)^n, 0; -1, 0]

The eigenvalues of this matrix are 0 and (-1)^n, which means that we have a center at (-1, nπ) when n is even, and a saddle point when n is odd. The linearization of the system near the center is:

x' = ky, y' = -kx

where k is a constant determined by the Jacobian matrix. The linearization near the saddle point is:

x' = -y, y' = -x

(b) To find the critical points of the system, we need to solve the equations:

1 - xy = 0 and x - y^3 = 0 for x and y, respectively. This gives us two critical points: (1, 1) and (-1, -1).

We can find the linearizations near each critical point by computing the Jacobian matrix:

J(x, y) = [-y, -x; 1, 3y^2]

At the critical point (1, 1), the Jacobian matrix becomes:

J(1, 1) = [-1, -1; 1, 3]

The eigenvalues of this matrix are -1 - √5 and -1 + √5, which means that we have a saddle point at (1, 1). The linearization of the system near the saddle point is:

x' = (-1 - √5)x - y, y' = x + (3 - √5)y

At the critical point (-1, -1), the Jacobian matrix becomes:

J(-1, -1) = [1, 1; 1, 3]

The eigenvalues of this matrix are 2 and 2, which means that we have a degenerate critical point at (-1, -1). The linearization of the system near the degenerate critical point is:

x' = y, y' = 2x + 2y

This system has infinitely many solutions, since the eigenvalues are equal.

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After solving the word problem, the large bag contains enough sugar to make 40 cakes

Since the large bag is 2.5 times larger than the small bag, and the small bag contains enough sugar to make 9 cakes, the large bag contains enough sugar to make:

2.5 * 9 = 22.5 cakes

However, since there are only 30 cups of sugar in the large bag, and we don't know how much sugar is needed to make a single cake, we cannot determine the exact number of cakes that can be made with a large bag of sugar.

As for the small bag, if it had enough sugar to make 9 cakes and there are 0.75 cups of sugar remaining, then each cake requires:

(sugar in bag - remaining sugar) / number of cakes

= (9 - 0.75) / 9

= 0.75 cups of sugar

Therefore, the large bag contains enough sugar to make:

30 cups / 0.75 cups per cake

= 40 cakes

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Uranus rotates on its axis at an angular speed of 0.355 radians per day, and a point on its equator travels at a linear speed of approximately 9,522 miles per day.

Uranus is one of the gas giants in our solar system, and it has a unique orientation compared to the other planets. Its axis of rotation is tilted at an angle of 97.77 degrees relative to its orbit around the Sun, which means that it essentially spins on its side. This also means that its equator is located in a plane perpendicular to its orbit, unlike Earth's equator, which is in the plane of its orbit.

Given that Uranus rotates on its axis once every 17.2 hours and its equator lies on a circle with a radius of 15,881 miles, we can calculate the angular and linear speed of a point on its equator.

Angular speed is a measure of the rate of change of an angle with respect to time. In this case, we want to know the angular speed of a point on Uranus' equator in radians per day. To find this, we can start by calculating the angle that a point on the equator travels in one day, which is equal to the angular speed times the time, or 2π radians (a full circle).

So, the angular speed of a point on Uranus' equator is:

(2π radians)/(24 hours) = 0.2618 radians per hour

To convert this to radians per day, we multiply by the number of hours in a day:

0.2618 radians/hour × 24 hours/day = 0.355 radians per day

Therefore, a point on Uranus' equator travels at an angular speed of 0.355 radians per day.

Linear speed is a measure of the rate of change of position with respect to time. In this case, we want to know the linear speed of a point on Uranus' equator in miles per day. To find this, we can use the formula:

Linear speed = angular speed × radius

Where the radius is the distance from the center of Uranus to a point on its equator, which we are given as 15,881 miles.

So, the linear speed of a point on Uranus' equator is:

0.355 radians/day × 15,881 miles = 9,521.9 miles per day

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The equation D = 2zv when solved for v is D/2z = v

To solve D = 2zv for v, we need to isolate the variable v on one side of the equation. We can do this by dividing both sides of the equation by 2z:

D = 2zv

So, we have

D/2z = v

So the solution for v is v = D/2z.

This equation tells us that v is equal to the ratio of D divided by 2z. In other words, v is proportional to D, and inversely proportional to 2z.

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Answer: A voting system

Step-by-step explanation:

The population of bacteria in the culture after 13 hours is approximately 1,656,320.

An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.

We have the initial population, Po = 65000, and the doubling time, d = 2 hours. To find the population after 13 hours, we need to use the formula:

[tex]Pt = Po * 2^{(t/d)[/tex]

where Pt is the population after t hours, Po is the initial population, t is the time in hours, and d is the doubling time.

Substituting the given values, we get:

Pt = 65000 x [tex]2^{(13/2)[/tex]

Pt ≈ 1,656,320

Rounding this to the nearest whole number, we get:

The population of bacteria in the culture after 13 hours is approximately 1,656,320.

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Total deposits of Sheri are $3,535.58 and Average check amount $160.32 and Change in balance is $3,055.61.

As the change in balance is positive, we can say that her overall balance increased as a result of these transactions.

To find the total of Sheri's deposits, we add up all the positive values in her checkbook:

Total deposits = $500 + $275 + $244.58 + $2,516 = $3,535.58

To find the average amount of her checks, we first need to find the total value of all her checks:

Total checks = -$13.52 - $15.88 - $451.57 = -$480.97

Then we can divide this total by the number of checks to get the average amount:

Average check amount = -$480.97 ÷ 3 = -$160.32

Note that the negative sign indicates that these are payments she made, rather than deposits.

To find the overall change in her balance, we add up all the entries in her checkbook:

Change in balance = -$13.52 - $15.88 + $500 - $451.57 + $275 + $244.58 + $2,516 = $3,055.61

Since this value is positive, we can say that her overall balance increased as a result of these transactions.

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An expression to represent the number of miles both boys ran this month is 3j + 4d + 10.

To begin solving this problem, we need to use the given information and create an expression to represent the number of miles both boys ran this month.
We know that James runs 3 miles per day, so in j days, he would have run 3j miles.

Similarly, Denis runs 4 miles per day and ran an additional 10 miles this month.

So in d days, he would have run 4d + 10 miles.
To find the total number of miles both boys ran this month, we need to add the number of miles James ran to the number of miles Denis ran.

Therefore, our expression is:
Total Miles = 3j + 4d + 10
This expression represents the total number of miles both boys ran this month.

To solve for j and d, we would need more information, such as the total number of miles the boys ran or the number of days they both ran.
In summary, we can use the given information about James and Denis's daily running habits to create an expression that represents the total number of miles both boys ran this month.

This expression is 3j + 4d + 10.

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8 ounces of chocolate are needed to make a gallon of chocolate milk. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

The four basic operations, also referred to as "arithmetic operations," are meant to explain all real numbers. Operations like division, multiplication, addition, and subtraction come before operations like quotient, product, sum, and difference in mathematics.

We are given that for making chocolate milk, in four cups of milk, 2 ounces of chocolate syrup is needed.

It is also given that 1 gallon = 16 cups

So, using multiplication operation gives

⇒ For 16 cups = 2 * 4

⇒ For 16 cups = 8 ounces

Hence, 8 ounces of chocolate are needed to make a gallon of chocolate milk.

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Recall that the exponential growth function is defined as f(x) = [tex]a^x[/tex], where a is a positive constant greater than 1. Since any real number can be plugged in for x, the domain of the function is all real numbers.

Since the exponential growth function increases without bound as x goes to infinity, the range of the function is all positive real numbers (y > 0). Similarly, as x approaches negative infinity, the function approaches zero but never equals zero. Therefore, the range of the function does not include zero or any negative numbers (y > 0).

So, the complete answer is:

Domain: all real numbers; Range: y > 0.

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All of the unit rates that are equivalent to the speed of the model airplane are 22 feet per second, 1320 feet per minute, and 0.25 miles per minute.

To calculate the speed of the model airplane, we need to use the formula:

Speed = Distance / Time

Given that the distance covered by the model airplane is 330 feet, and the time taken is 15 seconds, we can substitute these values in the above formula to get:

Speed = 330 feet / 15 seconds

Simplifying this, we get:

Speed = 22 feet per second

To convert this unit rate to other equivalent unit rates, we need to use conversion factors. For example, to convert feet per second to feet per minute, we can multiply the unit rate by 60 (since there are 60 seconds in a minute):

22 feet per second x 60 seconds per minute = 1320 feet per minute

Similarly, to convert feet per minute to miles per minute, we can use the conversion factor 1 mile = 5280 feet:

1320 feet per minute / 5280 feet per mile = 0.25 miles per minute

Therefore, all of the unit rates that are equivalent to the speed of the model airplane are 22 feet per second, 1320 feet per minute, and 0.25 miles per minute.

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The area under the catenary between the two poles is approximately 51.3224 square yards.

To find the area under the catenary between two poles 12 yards apart, with the equation y = 12cosh(x/12) - 5 between x = -6 and x = 6.

We can find the area by using integration.
The equation for the catenary.
y = 12cosh(x/12) - 5
Set up the integral to find the area under the curve between x = -6 and x = 6.
Area = ∫ (-6 to 6)[12cosh(x/12) - 5]dx
Integrate the function with respect to x.
Since we are dealing with the hyperbolic cosine function, we know that the integral of cosh(x/12) is 12sinh(x/12).

Therefore, the integral becomes:
Area = [12 (12sinh(x/12)) - 5x] evaluated from -6 to 6
Evaluate the integral at the bounds.
At x = 6: 12 (12sinh(6/12)) - 5(6) = 12 (12sinh(0.5)) - 30
At x = -6: 12 (12sinh(-6/12)) - 5(-6) = 12 (12sinh(-0.5)) + 30
Subtract the lower bound result from the upper bound result.
Area = [12(12sinh(0.5)) - 30] - [12(12sinh(-0.5)) + 30]
Calculate the numerical values and round to four decimal places.
Area = 51.3224

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Answer:6

Step-by-step explanation:3*4/2=6

Answer:

The range is 10, and the IQR is 13.

Step-by-step explanation:

The range is the difference between the maximum and minimum values in a dataset, and the interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1) of the dataset.

Since the range is the difference between the maximum and minimum values in the dataset, the range cannot be 14 and the IQR be 10, since 14 is greater than 10.

Therefore, the correct answer is:

The range is 10, and the IQR is 13.

Answer:it would still be a 50/50 chance of it be tails

Step-by-step explanation:

a coin has 2 sides. The probability would be 1/2. That means if you flip it a even amount, there would be a 50/tip chance. Let me know if I’m correct.

Where the above figures are given, The right triangles are: B and C.

Using the phythagoren theorem, we can determind the options that represent a right ttriangle.


A 5,8, and 9:

5^2 + 8^2 = 25 + 64 = 89

9^2 = 81

Not a right triangle.

B 20, 21, and 29:

20^2 + 21^2 = 400 + 441 = 841

29^2 = 841

It is a right triangle.

C 9, 12, and 15:

9^2 + 12^2 = 81 + 144 = 225

15^2 = 225

It is a right triangle.

D 5, 6, and 11:

5^2 + 6^2 = 25 + 36 = 61

11^2 = 121

Not a right triangle.

The right triangles are: B and C.

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a.) the equation for g(x) is g(x) = 1.1x + 1.1

b.)  The equation for g(x) is x = 10 which is the equation for a vertical line passing through (10, 15).

An equation for g(x) in point-slope form is:

y - 15 = m(x - 10)

We have that g(x) = f(x) = -11 and that g(x) passes through the point (-1, 0),

use point-slope form to write the equation for g(x):

y - 0 = (-11)(x - (-1))

We simplify and then solve for y

y = -11x - 11

In conclusion, the equation for g(x) in slope-intercept form is:

g(x) = -11x - 11

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To find dy/dx for y = 1/3x³ + 7x, we need to take the derivative with respect to x using the power rule and the sum rule, which gives dy/dx = x² + 7.

The given equation is y = 1/3x³ + 7x. To find dy/dx, we need to take the derivative of y with respect to x. We can do this by using the power rule and the sum rule of differentiation.

Using the power rule, the derivative of x³ is 3x². Using the sum rule, the derivative of 7x is 7. Therefore, the derivative of y with respect to x is:

dy/dx = d/dx (1/3x³ + 7x)

= d/dx (1/3x³) + d/dx (7x)

= (1/3) d/dx (x³) + 7 d/dx (x)

= (1/3) (3x²) + 7

= x² + 7

Hence, we have found that the derivative of y with respect to x is x² + 7.

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The probability of selecting an M&M that is not red is 11/15.To find the probability of selecting an M&M that is not red, we need to first find the total number of M&Ms in the bag,

It is the sum of the number of M&Ms of each color: 4 + 8 + 6 + 12 = 30.

Next, we need to find the number of M&Ms that are not red, which is the sum of the number of M&Ms of all other colors: 4 + 6 + 12 = 22.

Therefore, the probability of selecting an M&M that is not red is 22/30, which can be simplified by dividing both the numerator and the denominator by 2:

22/30 = 11/15

So the probability of selecting an M&M that is not red is 11/15.

In other words, there is an 11/15 chance that the selected M&M will be blue, orange, or green, and a 4/15 chance that it will be red.It is important to note that this assumes that each M&M is equally likely to be selected, and that the bag is well-mixed so that each M&M has an equal chance of being chosen.

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Answer:

To solve this trigonometric identity, we need to use the definitions of the trigonometric functions and some algebraic manipulation. Here's how we can do it:

cos 14° - sin 14°/ cos 14° + sin 14°

= (cos 14°/cos 14°) - (sin 14°/cos 14°)/(cos 14°/cos 14°) + (sin 14°/cos 14°) (multiplying the numerator and denominator of the second term by cos 14°)

= 1 - tan 14°/1 + tan 14° (using the definitions of cosine and sine, and dividing both terms by cos 14°)

= (1 - tan²14°)/(1 + tan 14°) (using the identity 1 + tan²θ = sec²θ)

= 1/cot 14° - cot 14° (using the definition of cotangent and simplifying the numerator)

= cot 90° - cot 14° (using the identity cot(90° - θ) = tan θ)

= cot (90° + 14°) (using the identity cot(θ + 90°) = -tan θ)

= cot 104°

Since cot(104°) = cot(180° - 76°) = -cot 76°, we can also write the final answer as -cot 76°.

Therefore, the given identity is true, and we have shown that:

cos 14° - sin 14°/ cos 14° + sin 14° = cot 59​ = -cot 76°.

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The standard deviation for the prices paid for this particular model of car is $6,250.

Based on the figure you provided, we know that the data is normally distributed and approximately 68% of car buyers pay within one standard deviation of the mean, approximately 95% pay within two standard deviations, and approximately 99.7% pay within three standard deviations.

Since we know that 95% of the prices paid fall within two standard deviations of the mean, we can say that the distance between the mean and the upper or lower limit of this range is equal to two standard deviations. This is also known as the "95% confidence interval."

Therefore, to find the standard deviation, we can calculate the distance between the mean and either the upper or lower limit of the 95% confidence interval and then divide it by two.

From the figure, we can see that the 95% confidence interval extends from approximately $23,500 to $48,500. The midpoint of this interval is approximately $36,000, which we can take as the mean.

So, the distance between the mean and either end of the 95% confidence interval is:

$48,500 - $36,000 = $12,500

Dividing this by two gives us the standard deviation:

$12,500 / 2 = $6,250

Therefore, the standard deviation for the prices paid for this particular model of car is $6,250.

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The standard form of the quadratic relation y=2(x-2)^2-18 is y=2x^2-8x-14.

The standard form of a quadratic relation is y=ax^2+bx+c, where a, b, and c are constants. To convert y=2(x-2)^2-18 to standard form, we need to expand the squared term and simplify the expression.

First, we expand the squared term to get y=2(x^2-4x+4)-18. Then, we distribute the 2 to get y=2x^2-8x+8-18. Finally, we simplify by combining like terms to get the standard form y=2x^2-8x-14.

This quadratic relation is a parabola with a vertex at (2,-18) and it opens upwards since the coefficient of x^2 is positive. The axis of symmetry is a vertical line passing through the vertex, and the y-intercept is -14.

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Weak correlation observed in scatter plot, inaccurate r=0.01 value; possible causal relationship - new fertilizer's effect on strawberry yields.

Based on the scatter plot, a correlation coefficient of r=0.01 is not an accurate value for this data. This is because the scatter plot shows an upward trend with moderately spread out points from the line of best fit, indicating a weak positive correlation. A correlation coefficient of 0.01 suggests a near-zero correlation, which is inconsistent with the observed pattern in the scatter plot.

A possible scenario for a causal relationship for strawberries picked on the farm could be the application of a new fertilizer that is known to increase strawberry yields. The farmer could divide the field in half, applying the new fertilizer to one half and the traditional fertilizer to the other half, and then compare the yields of each half. This would allow for a comparison of the effect of the two different fertilizers on strawberry yields and establish a causal relationship between the fertilizer and the yield of strawberries.

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With the given pay increase Daniel will earn a total of $185,141.90 in 4 years .

To find out how much Daniel will earn in 4 years with a starting salary of $42,000 and a 6.5% pay increase every year, follow these steps:

1. Calculate the annual salary for each year by applying the percentage increase.
2. Sum up the salaries for all 4 years.

Step 1: Calculate the annual salary for each year
Year 1: $42,000
Year 2: $42,000 * (1 + 6.5%) = $42,000 * 1.065 = $44,730
Year 3: $44,730 * (1 + 6.5%) = $44,730 * 1.065 = $47,656.95
Year 4: $47,656.95 * (1 + 6.5%) = $47,656.95 * 1.065 = $50,754.95

Step 2: Sum up the salaries for all 4 years
Total earnings = $42,000 + $44,730 + $47,656.95 + $50,754.95 = $185,141.90

Daniel will earn a total of $185,141.90 in 4 years with the given pay increase.

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The different teams of 6 students that can be formed for competitions is 3003

From the question, we have the following parameters that can be used in our computation:

Students = 14

Students in the team = 6

Using the above as a guide, we have the following:

n = 14

r = 6

The different teams of 6 students that can be formed for competitions is calculated as

Teams = nCr

substitute the known values in the above equation, so, we have the following representation

Teams = 14C6

So, we have

Teams = 14!/(6! * 8!)

Evaluate

Teams = 3003

Hence, the different teams of 6 students that can be formed for competitions is 3003

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